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Unformatted text preview: ﬁxed, each
a unique , we may have more than one as
varies but
there are ﬁnitely many such ’s for all ). , ¥
¦¥ £
PR
I
P
I
P I. Find
and all -way minimal global routings. The existence
of
and the ﬁniteness of the number of minimal -way
global routings are guaranteed by Lemma 1. , ¤ IP
¥ V H£
E
U¥ HP
¥
E £ I
I
PP ¥ ¨¤
¡ be a hyper-universal
L EMMA 2. Let
-design. Then
.
restricting on any two parts
gives a hyper-universal
-design, and restricting on any
three parts gives a hyper-universal
-design. The optimum
-design is a perfect matching, and an optimum
-design
is a Hamilton cycle. Moreover, the optimum
-design must be
a Hamiltonian cycle. In [8], we have developed a general reduction technique for designing
S-boxes. , 8 L EMMA 1. For any integer with
, there exists an integer
such that any -way global routing
could be decomposed
into minimal -way subglobal routings with densities at most
.
Moreover,
for
. , ¥
U¥ £
For Step III, [8] gave a hyper-universal
-design
with less than
switches.
Our goal in this paper is to further investigate Step III to obtain
better
-designs for
and and hence obtain a better
-design than the
-design
constructed in [8].
The following result was proved in [8] which will be used in this
paper. ¥£
U¥ ! Our approach depends on a very nice decomposition property
of global routings. Let
be a
-global routing and
be a sub-collection of
. If
is a
-global routing with
,
is called a sub-global routing o...
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